On a class of 3D second-order integrable Lagrangians and their dispersive deformations

[2608.15155] On a class of 3D second-order integrable Lagrangians and their dispersive deformations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Nonlinear Sciences > Exactly Solvable and Integrable Systems arXiv:2608.15155 (nlin) [Submitted on 15 Aug 2026] Title: On a class of 3D second-order integrable Lagrangians and their dispersive deformations Authors: Lingling Xue , E.V. Ferapontov , M.V. Pavlov View a PDF of the paper titled On a class of 3D second-order integrable Lagrangians and their dispersive deformations, by Lingling Xue and 2 other authors View PDF HTML (experimental) Abstract: We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also this http URL , dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively. Comments: 27 pages Subjects: Exactly Solvable and Integrable Systems (nlin.SI) ; Mathematical Physics (math-ph); Differential Geometry (math.DG) MSC classes: 35Q51, 37K05, 37K10, 37K20, 53D45 Cite as: arXiv:2608.15155 [nlin.SI] (or arXiv:2608.15155v1 [nlin.SI] for this version) https://doi.org/10.48550/arXiv.2608.15155 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Lingling Xue [ view email ] [v1] Sat, 15 Aug 2026 10:12:26 UTC (36 KB) Full-text links: Access Paper: View a PDF of the paper titled On a class of 3D second-order integrable Lagrangians and their dispersive deformations, by Lingling Xue and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: nlin.SI < prev | next > new | recent | 2026-08 Change to browse by: math math-ph math.DG math.MP nlin References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• On a class of 3D second-order integrable Lagrangians and their dispersive deformations#611104
• Auxiliary Field Deformations of the Lambda Model#1012990
• The Euler–Poisson equations; an elementary approach to partial integrability conditions, Goryachev–Chaplygin and beyond#630487
• Integrable Elasticity via Neural Demand Potentials#216807
• Riemannian L-systems: Modeling growing forms in curved spaces#464298